Seminarium KMMF "Teoria Dwoistości"

I will discuss "motivic" invariants of complex algebraic manifolds. Such invariants satisfy the so-called scissor relation, just like the usual topological Euler characteristic. Motivic invariants are well defined on the Grothendieck ring of complex varieties and can be generalized to the singular spaces. Among motivic invariants which are expressed by Chern classes there exists a universal one. This is the Hirzebruch χy-genus. In a situation when the multiplicative group |C* acts on an algebraic variety, we apply localization theory. We express the global χy-genus in terms of local contributions coming from the fixed-point set. The resulting decomposition of the χy-genus specializes to the decomposition arising from the Białynicki-Birula plus-decomposition.